Singular peturbation problems in ship hydrodynamics

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a

In Memory: Reinier Timman, 1917-1975

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91

This article was written while the author was Visiting Professor in the Department of Naval Architecture, Osaka University, Japan, and in the Mathematics Department, Man-.*, clIester University, England. These visits were supported partially by grants from the Japan

Society for the Promotion of Science and the North Atlantic Treaty Organization, respectively. Leboratorium veer Scheepshydromeehenl=

Archlet

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Singular-Perturbation

Problems in Ship Hydrodynamicsi

T. FRANCIS OGILVIE

Department of Naval Architecture and Marine Engineering The University of Michigan

Ann Arbor, Michigan

I. Introduction

II. Slender-Body Theory in Aerodynamics

92

95

111. Slender Ships in Unsteady Motion at Zero Speed 105

A. Problem Formulation 105

B. Radiation Patterns 107

C. Forced Oscillations 111

D. Diffraction Problems 125

IV. Slender Ships in Steady Forward Motion 145

A. Introduction 145

B. Ordinary Slender-Ship Theory 147

C. Failure of Ordinary Slender-Ship Theory 151

D. The Bow-Flow Problem 156

E. The Low-Speed Problem 166

V. Slender Ships in Unsteady Forward Motion 169

A. Introduction and Formulation 169

B. Solution of the Heave/Pitch Forced-Oscillation Problem 176

C. High-Speed Slender-Ship Theory 182

References 185

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.. . . . .. . .. . . . . .

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92

Introduction

Michell (1898), laid thefoundation for modern ship hydrodynamics in his 'extraordinary paper on the wave resistance of a thin ship.Havelock 1(1923) 'discovered and extended Michell's work, and for 40 years thereafter progress in ship hydrodynamics was almost synonymous with development of thin-ship theory. Havelock pointed out in his first paper on this subject that one may question the validity of thin-ship theory in practical applicationssince real ships are hardly thin.Nevertheless, study of the thinship prospered, and the theory provided someguidance in the reduction of wave resistance,, even if it did not give accurate quantitative predictions.

The early and continuing success of thin-ship theory was due to two

factors: i(i) An explicit solution can be found for the case of a ship in steady Straight-ahead motion. (ii) The first approximation is essentially "a uniform

approximation.

Khaskind (1946) and Peters and Stoker (1957) also obtained an explicit solution for a heaving and pitching thin ship with forward motion. In all such problems, the ship is replaced by a centerplanedistribution of sources, steady or pulsating, as appropriate. The body boundary condition is trans-ferred to the centerplane, thus providing an 'expression for the normal ve-locity component on each side of the centerplane,,from which an explicit expression for source density can be obtained. If the potential for each source in thedistribution satisfies the free-surfacecondition and a radiation condition, the potential for the superposition of sources satisfies approxi

mately all of the conditions of the problem. Then itis a straightforwarc

matter to find the pressure and to integrate it appropriatelyover the hull tc obtain the force on the ship. Alternatively, the momentumtheorem or at .energy-flux theorem can be used to obtain similarresults from the solutioi at great distance from the ship. [The latter can evenbe used to obtain som

of the desired quantities in the unsteady-motion problem,, as shown b

Newman (1959).]

If the ship is not symmetrical, either because of itsactual shape ox becaus it is yawed, this procedure does not work. A distribution of transverse d poles must also be placed on the centerplane in thecorresponding mathemat

cal problem, and an integral equation must be solved to determine th density of the dipoles. This is a two-dimensional (2-D) singular integn

equation with a very complicated kernel, and only the adventof the moder high-speed digital computer made its solution possible. This was first dor by Daoud (1973).

Similarly, if one tries to satisfythe boundary conditionexactly on the trt hull surface instead of on the 'centerplane,, a 'complicated,integral equatic

V

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must be solved, and there is still some controversy about the proper way to do this.

Nevertheless, the thin-ship model (perhaps with modifications)still pro-vides the best strictly analytical information availableabout the wave

resis-tance of a ship,

largely because an explicit solution is obtainable.

Corresponding results for ship-motion problems have not been so useful, at least partly because a better mathematical model has evolved. (This other model will be discussed in this article, but we may note that it has not been

successful in the steady-motion problem.)

The thin-ship solution is the first term in a perturbationexpansion of the corresponding exact problem. The small parameter may be considered as the ratio beam/length. As this parameter, say, /3, approaches zero,the ship reduces to just its centerplane projection, and the hull boundary condition must be satisfied on the two sides of this surface. If, on theother hand, one formulates a slender-ship problem, in which both beam and draft approach zero with the small parameter,say, E. the hullshrinks down to a line, and the limit problem is not a well-posed problem in potential theory. Sincethe limit problem cannot even be posedproperly, let alone solved, one cannot obtain a first approximation to the solution in this way.Thus, in spite of the fact

that a ship appears to be more"slender than thin," the mathematician was forced until recently to consider it only as thin.

The slender-ship problem is a singular-perturbation problem of a kind that aerodynamicists have learned to solve in the lastseveral decades (see,

for example, Ward, 1955).

Thus it was to be expected

that similar approaches should have been made to ship problems in recent years.

However, when the aerodynamic techniques were applied in a

straight-forward way to ship problems, the results were generally disappointing,

being sometimes trivial, sometimes nonsensical. Only in the last 10 years have these techniques been satisfactorily adapted to the specialconditions of ship problems so as to lead to reasonable and sometimes useful results.

The treatment of these singular-perturbation problems is thesubject of this article. The concept of the

slender shipand all that that

implies for

ship hydrodynamicsshould notbe considered as competition or a

replace-ment for the thin-ship concept. Rather, it should be considered as a comple-ment to thin-ship theory, offering some new understandingof the physical phenomena of ship hydrodynamics and perhaps eventually leading to some

alternative methods of calculating quantities of practical interest.

There is a growing body of investigators in ship hydrodynamicsworking on slender-ship theory. This article is not intended primarily for them, but rather for two other groups: (1) those who are still workinglargely with the older, more established concepts. who wish to learn quickly theprinciples, techniques, and possibilities of slender-ship theory, and (2) people who are

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94

familiar withsingular-perturbation techniquesin other areas, especially with slender-body theory ofaerodynamics, who are interested in finding outwhat is being done with these techniques in shiphydrodynamics. The section on slender-body theory in aerodynamics is provided as abrief introduction for

the first group only.

There are some singular-perturbation problemsof ship hydrodynamics that are neglected. largely because theydid not fit in with thegeneral theme as we have organized it. In this category, wemention especially the work by Rispin (1967) and Wu (1967), in which they show how to solve the 2-D planing problem by matching a locally nonlinear description (without

grav-ity in the first approximation) to a far-field gravity-wave solution. In a

similar way, Shen and Ogilvie (1972) developed atheory for a high-aspect-ratio planing surface in which the local picture is a 2-D nonlinear flow,just like the near-field solution of Rispin and ofWu, while the far-field solution is a lifting-line approximation.

There has also been considerable interest in recentyears in high-frequency problems and in low-speed, steady-motion problems, in both of which the waves characteristically have a very small wavelength, and so the wave motion is confined to a thin layer near the freesurface. These problems are barely mentioned in Sections III,C,3 and III,D,5.From a scientific pointof view, they are very interesting assingular-perturbation problems,since they are truly boundary-layer problems. However, some choices had to be made, and the theory for such problems is not discussed in any detail.

Also, as implied in themention of" potential theory," we assume that the fluid is ideal. Thus, problems of a viscousfluid are not considered at all. Of course, classical boundary-layer theory is one of the best-known of all singular-perturbation problemsin fluid mechanics,and an understanding of viscous boundary layers is of critical importance in ship hydrodynamics. The boundary layers on ships are, however, always turbulent, and so the mathematical theory ofboundary layers is less useful in ship hydrodynamics

than in some other branches of fluid mechanics. To have included such

problems in this article would have required the introduction of a tremen-dous and largely independent subject, probably with marginal benefit, and so there will be no further discussion of nonidcal fluids.

In general. we have tried to emphasize the physical understanding of problems and, where possible, to indicate thelines of thought thathave led to the various formulations and solutions. The research worker first gets an idea that something may possibly be true, then tries to find out whether it really is true, and, if he convinces himself, he then tries towork out a proof (or performs experiments) toconvince others. It is unfortunate that only the last of these stages is usually reported, and wehave taken the conscious risk of trying to report something from the first twostages as well.

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In this respect, much of the work discussed here (especially our own work)

is based on the method of matched asymptotic expansions. We prefer to

think about thee problems along much the same lines as Van Dyke (1975). Now, Van Dyke's book is perhaps unsatisfactory tomathematicians, but we believe that he provides an extremely valuable way of thinkingabout

physi-cal problems, a way that

often suggests practical solutions. Thus his

approach is a useful part of the

first stage of scientific investigation as

described above. By itself, of course, it proves nothing, and a lot of nonsense

has been "derived" with the help of the method ofmatched asymptotic

expansions.

In providing references, we have tried to give those that might be of interest for further reading, rather than trying to establish priority of publi-cation or presenting a comprehensive bibliography of the subject. To those who thereby feel neglected, we can only apologize and ask their

understand-ing; we did not want to produce an annotated bibliography, which this article would surely have become had we tried to include all pertinent

references.

In matters of notation, we consistently follow the usage of Abramowitz and Stegun (1964).

II. Slender-Body Theory in Aerodynamics

The basic concept of the slender body in aerodynamics originated with Munk (1924). He was concerned with the flow around an airship. He noted

that such a body has one long

axis, parallel to which the fluid velocity

component is generally much smaller than the components in the

cross-planes. If the reference frame is fixed to the fluid at infinity, one may imagine that the forward motion of the slender body causes the fluid to

part--to move sidewayswithout

being greatly disturbed in the

longitu-dinal direction. There is approximately a 2-D flow in each crossplane. Thus

the 3-D problem is greatly simplified in being replaced by a set of 2-D problems, and furthermore the powerful methods for solving 2-D flows

become available.

Of course, such a viewpoint is only valid rather close to the body. Very far away, the flow field caused by any moving nonlifting body appears as if it might have been created by a moving dipole, and such a flow field is in direct

contrast to the notion of a 2-D flow pattern. Thus, it must be recognized

from the outset that the slender-body description of the flow field is not valid uniformly far away from the body.

Furthermore, there generally are stagnation points at the nose and tail of the body, at which points the fluid velocity is approximately equal to the

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96

velocity of the body itself, ie., the longitudinal fluid velocity componen dominates the transversecomponents. At such points, the slender-body con cept is completely wrong.

Notwithstanding these twodifficulties, it has been found that prediction of slender-body theory are fairly good in many important aerodynami problems, and so the theory has beenstudied and furtherdeveloped over period of severaldecades. Starting in the 1940s especially, when many

acre

dynamic vehicles became effectively slender, basic understanding c slender-body theoryimproved, the accuracy of its predictions

increased, an the scope of problems to which it could be applied widenedgreatly.

The final stage in the development of aerodynamic slender-body forma lism occurred in the 1960s, when the methodof matched asymptotic expar sions was applied to such problems. A good textbook treatment has bee provided by Ashley and Landahl (1965). This method has not receive universal acceptance by workers in the field, but its practical usefulnes cannot be denied, and it will be used here. In particular, it provides som simplified viewpoints on slender-body problems and it also provides formal procedure for relating those viewpoints mathematically. One ma

also comment that itgives the illusion of proving various conclusions,and it can be dangerous when used improperly.

In slender-body theory, we infer the existence of a small parameter

which is a measure of the "slenderness" of the body under study. Thi

parameter may be simply the ratio of a typical transverse dimension, sa diameter, to the length, or it may be the maximumslope with respect to th longitudinal axis of planes tangent to the body surface. For developing ti' general theory, its precise definition does not matter. What does matter that the entire theory is developed in an asymptotic sense as s

0. That i any conclusions will be more nearlyvalid as E becomes smaller and small( Whether such conclusions are valid for any finite value of E will always be matter for further investigation in any particular problem.

For convenience in discussing slender-bodyproblems of all kinds, we sln generally consider that L, the body length, is0(1)as E 0, i.e., L is fixed ai its value is not affectedby the limit operation. Then, a transversedimensi, can be described as being OW as E 0. If any ambiguity is likely to ar from such usage, we can revert to the moreusual statement, B/L = 0( where B is the transverse dimension of interest.

The basic concept ofslender-body theory can be stated mathematically follows: Iff (x, y, z) is any flow variable, then, in someregion near the bo,

aflay, eflaz = 0(fle), Of/ax O(f), as e 0, (:

where the x axis is chosen as the longitudinal axis of the body. This sta ---0

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merit says simply that derivatives in the transverse directions are very large compared to derivatives in the longitudinal direction. The region in which these statements are assumed to be valid has transverse dimensions that are 0(e). These statements are assumptions, and so they cannot be proven.

However, they can be made to appear plausible, as is suggested in the

following two typical situations:

(1) In Fig. 1 there is depicted a section of aslender body which is moving

at speed U in

the longitudinal direction through fluid at rest. Let V = (v1, v2, v3) be the fluid velocity caused by the motion of this body. At

V2

FIG. I. A slender body in steady longitudinal motion.

points A and B, the v2 component has almost the same magnitude but

opposite signs. Thus At', defined as v, IA - v2113, has avalue comparable to

twice v2 IA. The distance AB between the two

points is 0(c), and so

Av2 /AB = 0(v2 /e). It follows that the derivative of v2 along a path connect-ing points A and B must be 0(v2 /e), and thus that derivatives in the trans-verse planes are large. On the other hand. there is no such reason to expect that 8v2lax should be large.

(2) In Fig. 2 there is depicted a cross section of a slender body moving with speed V in the transverse direction. The velocity component v, at point A must equal V. but at point B 12 is very nearly equal to V. Again, the distance AB is 0(c), and so the same argument leads to the conclusion that

2/13Y = 0(v2/c).

Such arguments are not so clear in the case of some flow variables, but nevertheless the relationships indicated in (2.1) can generally be justified,

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-98

FIG. 2. A slender body movingin the transverse direction.

especially if one is willing to overlook the difference in asymptotic behavior between, say, OW) and 0(v" log Er

In order to use the assumption stated in (2.1), let us consider a slender

body moving with velocity U = (U5, U2, U3)(possibly a function of time).

We assume that the fluid motion can be described in terms of a velocity

potential, 4)(x, y, z, t), which satisfies the Laplace equation

in three

dimensions,

\720 =+ 0 + Ozz

=0.

(2.2)

0(0) 0(41/v2)

The orders of magnitude of the terms in (2.2) have been noted. Although we have not yet determinedthe dependence of (/) on e, wecan assert that thefirst term in (2.2) is of higher order than the other two terms, and so it can be neglected in the determination of the first approximation for 0. The poten-tial also satisfies a kinematic body boundary condition,

430/an = n U, (2.3)

or

n,cp + n2(f)y+ n3( z=-- n,U, + n2 U2 + n3 U3 , (2.3')

0(cbv) 0(4)/v)

Of course, the asymptoticbehavior of the two is not the same. Nevertheless, 1 shall treat

them as if they were. This never seems to lead totrouble, and there are two verypractical reasons for doing so: (i) A function such as log .4E can always be broken into two terms,

log .4 + log E, and, if A =0(1) as e 0, these two terms are 0(1)and 0(log E), respectively. However, the value of A can be changed by making a new definition fore, and so it is clear that

the two terms cannot betreated independently, even if they are formally of different

orders of magnitude with respect to E. (ii) We derive asymptotic formulaswith the intention of applyin them for finite values of L.although the formulas are onlyvalid asymptotically as e 0. If wt

had to restrict the rangeof e so that the differencebetween log e and I were large, we woult probably never obtain a usefulformula.

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where

n = (n1, n2, n3) (2.4)

is a unit vector normal to the body surface, directed into the body. For a

slender body, it is important to note that

= 0(E) and n2, n3 0(1) as & O. (2.4')

This fact has been used already in noting the orders ofmagnitude of the terms in (2.3'). Also, it has been assumed in (2.3') that U, = 0(1) and that U2, U3 = 0((;), which are useful choices (if arbitrary). These are the largest possible orders of magnitude for the Ui's which still allow small distur-bances" to result. Finally, (2.1) has been used in estimating theleft-hand side of (2.3'). It is evident that we can drop n, 4 from (2.3'), at least for the first approximation, and we then find that

= 0(&2), (2.5)

which is a typical result of slender-body theory.

One further approximation, consistent with the above, is usually made. The left-hand side of (2.3) is a directional derivative of 4) taken along the normal vector n. On a slender body, this vector is almost perpendicular to the longitudinal axis. In finding the first approximation for 4), we can replace 00/Thi by 4/ON, where N is a unit vector,lying in a crossplane, perpendicu-lar to the contour of the body in that crossplane. The relative error incurred in this replacement is 0(c2).

Thus the first approximation for (strictly, the first termin an asymptotic

expansion) satisfies the following conditions:

45yy + = 0; (2.6)

aoloN=

N24 ,+ n1 U1 + n2U24- n3 U3. (2.7) This is strictly a set of 2-D problems, as was anticipated. The solutions can be found analytically or numerically by standard methods. Apparently the solution at each crossplane is independent of the solution at all other

cross-planes, except insofar as they are related by the body geometry, which

enters into (2.7). But in fact this is not true, for the simple 2-Dsolutions in the crossplanes are not unique: At any x,the solution of (2.6) which satisfies

(2.7) is arbitrary to the extent that any constant may be added without

affecting the validity of the solution. This constant may be different at

var-ious crossplanes, and so we must add an arbitrary function of x to any

solution determined from (2.6) and (2.7). We must somehow determine this

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100

additive function of x, and it turns out to represent a further interaction among the various crossplanes.

The additive function

of x cannot be

determined from the near-field analysis that we have been considering. We must construct a far-field

description of the flow in order to determine the interaction among the crossplanes. Before doing this, we should note how the approximate solution of the above near-field problem behaves for large r =(y2 + z2)". Since 0 must be harmonic everywhere outside the body contour, and its gradient must vanish at infinity, can berepresented as the real part of a complex potential, which can in turn be represented by a Laurent series supple-mented by a logarithm term. The potential 0 itself then takes theform

0(x, j7, z; t) = [o-(x; t)/27z] log r + Ao(x; t)

A(x; t) cos nO + B(x; t) sin tz0

+ E

rn (2.8)

n= 1

where 0 = arctan y/x. Since 0 -= 0(c2) and (in the near field) r = 0(c), we may conclude that

a =

0(c2) and

A,

= 0(vn+2). (2.8')/

At very large r, the first two terms in (2.8) dominate the others, and so

0(x, y, z; t) [o-(x;

t)/27r] log r + Ao(x; t) as r oo. (2.8")

Thus, very far away (but still in the near field),the flow caused by the body appears to have been caused by a 2-D sourceat the origin. Wenote that the strength of the source can be computed from (2.7), (2.8), and the equation of continuity, with the result that

c(x; t) = U ddS(x)/dx], (2.9)

where S(x) is the cross-sectional area of thebody.

From the far-field pointof view, as c --+ 0 thebody shrinks down to a line, the segment of the xaxis between x 0 and x =- L. It isthis fact that makes

slender-body theory a singular-perturbation problem even

in the first

approximation, for it is not a well-posed problem in 3-Dpotential theory to prescribe boundaryconditions along a line. If we require that the potential

function be harmonic everywhere other than on this line and that it

be bounded at infinity as well, then 0 must besingular on the fine segment. A convenient way to obtain ageneral form of the far-field solution is to restate the problem in terms of Fourier transforms with respect to x; the resulting 2-D problems can besolved in a general wayand then transformedagain tc

We are ignoring differences that are 0(log c). See the footnote on p. 98. =

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where

d)(x, 3i, z;,t.) = dke"

tr)[a:(k; t),cos b:(k; t) sin. nOr,

n=0

' (2.10) 'where the functions a(k; t) and b:(k; t) are 'unknown functions that can

themselves be interpreted as Fourier transforms). Near the x axis, which includes the singular line, r is small, and the terms in (2.10) can be approx,, imated by using the small-r series expressions for the Bessel functions

r

dkeikx K0( iktr)a(k; t) 1 _{21ft log r}

dkvat(k;

2n .- \j'r, OD

;

1 2"-

1Y! cos

nO dk - eikxc(k; t) n >0.

(.1 lb)

24Ern sin P_ Ilk In

In (2.1 ia) we see that the first term on the right-hand side behaves like log r. in a crosspiane, and so it appears to represdnt a 2,-D source of strength

alx; = 2ncto(x; t)/ (2.12a)

1

no(x; t')`-= dkel"dt,(k;

The full 3-D expression on the left:hand side of(2.11a) can also be rewritten 'to show that it represents a distribution of sources in three dimensions

21n f

dke'"-Wo(ik ir)at(k; t)=

r"

Ei(; t) (21213)

oo 4n . _co [(x

The second term on the right-hand side 01(2.1 la) is simply a function of x; 'using (2,12a), we can rewrite this function as follows

CI k

dkexa(k; t) log:

2 .CD (; t) log 211x sgrr(x

f(x;

(2.13) n

t All special functions are denoted as by Abffrriowitz and Stegtm 09641 Here, K. is thee modified Besse function of he second kind.

:log C 7 is the Euler constant.

Ship Hydrodynamics t

give.solutions of the form

y-.11 dkelkxat (k; Wog

Oki

2

t . COS

1

dkekx1c( ikir)c:(k; t)

nO.

-00 sin 2n . )2

r2rt

(2.11a)t4 k nO + t) -'(n - t). I = t),

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102

where the prime denotes differentiation with respect to (;. The left- and

right-hand sides of (2.11b) can also be interpreted in terms of3-D and 2-D distributions of multipoles. It is important to recognize that there is no simple function of just x left on the right-hand side of (2.1lb).

It is thus seen that the generalfar-field solution can be expressed in terms of distributions of sources, dipoles, etc., along the x axis. The strengths of these singularities are of course not known. From (2.12b), we might infer that the distribution extends along the entire x axis, but in fact it must be limited essentially to the line segment that representsthe remnant of the body in the limit z 0; we can simply set "d(x; t) = 0 outside of this segment. A similar step follows also for the multipole distributions.

Near the line of sources, the potential corresponding tojust the sources takes the following approximate form

(1/2706-(x; t) log r + f(x; t); (2.14)

this is the right-hand side of(2.11a) after we use (2.12a) and (2.13). A com-parison with (2.8") indicatesthat this is precisely the kind ofbehavioi- of the far-field potential that we would have liked to find. It followsdirectly that

6-(x; t) = o-(x; t),

the value of which is known from (2.9). The expression in (2.8") represents the near-field potential in its limiting behavior for large r; the expression in (2.14) comes from one term of the far-field potential, giving the limiting behavior of that term for small r. This is the kind of" matching" that forms

the basis for the method ofmatched asymptotic expansions. It has been a

well-known procedure ever since Prandtl first developed classical

boundary-layer theory, although much of the formalism has been developed only

recently.

This matching does not appear to be perfect yet, for we have considered just one term in the series in (2.10).The subsequent terms are more and more singular near the x axis, as (2.11b) shows; it appears that the terms with ii = 1 should dominate the n = 0 term if r is small enough, and the n = 2 terms should dominate the 17 = 1 terms, etc. If this were true, the matching would be destroyed. Physically it does not make sense either; if we start very close to the body, where the flow appears as if it might have been generated by sources, dipoles, quadrupoles, and so on, and then we move farther away, the sourcelike behavior decreases most slowly. Then, if we start very far away and approach the body, the sourcelike behavior should appear first.

We conclude that in the far field the lowest-order term in the solution

expansion should be sourcelike; other terms areof higher order with respect to E, that is, a: and b* are o(4) as E 0 for n > 0. Then the first term in the

.sansuM9721...121110.0

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-outer expansion is

(x, y, z, t)

dke'K0(lkir)4(k; t)

(2.10')

2ir _0,

t) log r + f(x; t) as r 0. (2.14')

2ir

Now the one-term outer expansion is completely known,since 5-, and thus ao, is known. Also, the one-term inner expansion is known, including the additive function of x, for Ao in (2.8") must be the same asf(x; t), given by

(2.13).

In the aerodynamic problem, one can proceed to obtain higher-order

solutions, although "end effects" are likely to cast doubt on the validity of such extended expansions. In any case, in the free-surface problems that are the focus of this article, it is difficult enough to obtain afirst approximation, and so we are not likely to be concerned with anythingbeyond. Only a few comments on the interpretation of the above results need yet to be made.

The additive function in the near-field 2-D solution, Ao orfhas a simple interpretation. At any x it represents an axial flow that is needed to correct the 2-D result represented by the log r term in the approximatesolution. To see this, one can start with the right-hand side of(2.12b), which is genuinely three-dimensional in nature. Integrate it once by parts with respect to the result is

1

27ru(x; t) log r

1- cgav; t) log{ I x + [(x )2 + sgn(x

4Th

As r 0, the second term hereapproaches a well-defined limit, which in.fact is equal to f (x; t). If the problem really were two-dimensional, sothat 6- were

a constant, this extra term would vanish. But the problem is not two-dimensional, and f(x; t) represents the value on the axis of the potential

which corrects for the nonzero longitudinal rate of change of

The 2-D solutions in the various crossplanes are related in two ways:(i) through the body geometry, which enters into the boundarycondition on the body at every cross section, and (ii) through theadditive "constant" at each section, f (x; t). The latter depends only on the sourcelike behavior of the flow in the cross-sectional planes, and so it vanishes if thereis no source-like behavior. There are two cases in which this happens.

(I) The body has no thickness. Although this cannot happen in reality, we often assume that a body has no thickness if we are interested primarily in

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104

the effect of the body as a lifting surface. In this case, in the near-fielc

solution as given in (2.8), the source term vanishes, and the lowest-orda term in the solution expansion is a dipole term. In the far field, we must ther

start the solution with a line distribution of dipoles, and, as seen frorr (2.11b), there is no function of x alone when this singular potential i!

estimated near the line of singularities.So. Ao in (2.8) vanishes. Slender-bod.1 theory for the lifting body of zerothickness reduces to the simplestkind of strip theory, with no interaction amongthe cross-sectional flows.

Even if we consider that a lifting body does have thickness, the interac. tions among sections depends only on the thickness effects.

(2) There is no forward 'notion. In the body boundary condition (2.7), le = 0. Again, the source strength is zero. This is indicated by (2.9), but i could also have been deduced from the fact that the 2-D problems represen a translating body, which has no sourcelike behavior. As in the case of th( lifting body of zero thickness, all source terms vanish and the interactior term Ao (or f) vanishes with them. The slender-body theory for vertical ol lateral oscillations of a body at zero speed is thus a primitive strip theory

This reduced capability of slender-body theory to represent interaction: among cross sections leads to a special difficulty in the case of slender liftim surfaces: It is generally impossible to satisfy a Kutta condition. A cros: section forward of the trailing edge isunaffected (in the first approximation by what happens at the trailing edge, and so in the theory there can be n( adjustment of the pressure and velocity fields ahead of the trailing edge ti ensure a smooth flow from the edge.

This difficulty has been rectified in recent years by the use of an accelera tion potential. See especially Newman and Wu (1973), who treat the prob lem of unsteady motion of a slender lifting body with thickness by such method. Also, Rogallo (1969) has shown the nature of the failure of lifting surface theory near the trailing edge of a slender wing.

More generally, errors at the body ends cause the worst problems i

aerodynamic slender-body theory. The basic assumptions are violated i such regions, even without difficulties in satisfying a Kutta condition. Sin

ilarly in applications to ship hydrodynamics, the theory is most limitc

in describing the fluid motion near the ends. The nature of the end effects quite different in the two fields, however,largely because of the presence 1 the free surface in ship problems, and little benefit would be gained from discussion here of problems that arespecifically aerodynamic.

As a final comment about aerodynamic slender-body theory, we shou mention how the pressure computation is affected by slenderness. In ord to keep the results concise, let us suppose that, if there is a steadyforwa: motion, we introduce instead an opposite uniform stream at infinity. TI

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pressure is computed from the Bernoulli equation,

P/P = tRi5. + 0,2 +

O(E2)

0(0)

o(c2)

The orders of magnitude are noted below the terms, the estimates being based on (2.5). We have assumed that 4, = OM. Thus, in the first-order

approximation for the pressure, we have

P/P =

Ot 1(0,2 +

= (NE2). (2.15)

The appearance here of two of the quadratic terms is a characteristic of

slender-body theory. Even if the velocity-potential problem itself can be treated as a linear problem, a consistent computation of the pressure gen-erally requires that the two quadratic terms be included.

A partial exception will be found later when we develop the solution of some motion problems in terms of two small parameters, in which ease we shall neglect quantities that are quadratic in terms of the second parameter. However, the argument will still stand with respect to quantities quadratic in terms of the slenderness parameter e.

III. Slender Ships in Unsteady Motion at Zero Speed

A. PROBLEM FORMULATION

Now we consider a slender body whose longitudinal axis is approximately parallel to the undisturbed free surface. Since we have it in mind to apply the results to ship problems, we assume that the body is partially submerged, that is, it intersects the free surface. The body may oscillate vertically or horizontally, it may be fixed and exposed to incident waves, or it may be free to respond to incident waves. We shall not consider the case of longitudinal oscillation of the body (surge, in nautical terms), largely because it is

ill-suited for the application of slender-body theory; we do not believe that

anyone has yet shown whether it is valid to simplify this case through the use of the slenderness property, although such an analysis has been performed formally.

It will be assumed throughout this section that the ship has no forward speed. In the notation of Eq. (2.3') or (2.7), we have U, = 0. Because of the presence of the free surface, the resulting theory is not necessarily a primitive

strip theory, as was always the case in the corresponding aerodynamic

problem.

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106

convenient at many points to assume that the unsteady motionis sinusoidal.

If we are concerned with just the linear problem, other motions can be

studied as superpositions of sinusoidal motions, and there is no loss of generality in restricting ourselves to sinusoidal motions.

It is still assumed that the body geometry allows for the definition of a slenderness parameter r with implications essentially the same as in the infinite-fluid problem. Again wetreat the fluid-motion problem as a problem in potential theory, and so the governingpartial differential equationis the Laplace equation in three dimensions. The kinematiccondition on the body is the same as before. Thus we start out requiring that (2.2) and (2.3) be

satisfied, but with U = 0 in the latter. Also, the arguments that were pre-sented with respect toFigs. 1 and 2 are still valid, and so we still accept (2.1)

as a reasonable hypothesis in the near field, although we shall have to

introduce some modifications presently.

The new aspect is the presence of the free surface. Theboundary condi-tions to be satisfied there arewell known [see, for example, Stoker (1957) or Wehausen and Laitone (1960)1. On the free surface, z = y, t), the pres-sure is constant, and we can set it equal to zero in the Bernoulli equation

pl p = -

(P, -

-

10! +

+ =0 on z C(x, y, t). (3.1)

There is a kinematic condition to be satisfied on the free surface

OxCx+ (I)Z =0 on z -= 1;(x, y, t). (3.2)

In addition, we must satisfy a radiation condition: we postpone specifying this condition until we have a linear problem to work with.

We want to simplify the free-surface boundary conditions in a way con-sistent with the concepts of slender-body theory, asdescribed in Section II. In the near field, we initially apply (2.1) as before.We must also consider what effect differentiation with respect to timehas on orders of magnitude; this is necessary if we are to interpret the time derivativesin (3.1) and (3.2) correctly. There is a rangeof possible choices, but only three of these lead tc significant results:

(i) 4/51 = 0(0); (ii)

04)/ät = 0(çb/012); (iii) 0010t =

0(4)10. It will be necessary to provide an acceptable interpretation fol presuming that time differentiation is related (in an order-of-magnituch

sense) to the slenderness parameter E, and we shall do this presently. It will also be necessary to satisfy (3.1) and (3.2) in the far field. Corre sponding to the three possibilities just mentioned for theorder-of-magnitud,

effect of the operator 0104, we shall find that some modifications of th

infinite-fluid ideas are needed in the far field.

Before examining these cases separately, it is worthwhile to anticipat some simple results and to use them to provide some physical insight int the meaning of the several choices. That is the purpose of the next sectioi

C(x, =

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B. RADIATION PATTERNS

In the far field, it turns out that the most reasonablefree-surface condition

is

4) + gOz = 0 on z = 0. (3.3)

This will be discussed further in Section III,C. For now, we note only that it is easily obtained from (3.1) and (3.2) by retaining just the linear terms and eliminating between the two equations. Equation (3.3) is the classical free-surface condition for linearized problems involving unsteady motion of a fluid with a free surface. We shall show later that (3.3) should be used in the far field regardless of what assumption is madeabout the effect of the

opera-toralat.

Consider the case of a ship heaving (translating in thevertical direction). If the ship is sufficiently slender, the disturbance far away appears as if it might have been created by a distribution of time-dependent sources onthe x axis. This is somewhat analogous to the result for a slender body in an infinite fluid, as expressed in (2.10), but there are two important differences

to be noted: (i) In the present case, the simple sources in (2.10') must be

replaced by free-surface sources, that is, the corresponding velocity poten-tials must satisfy (3.3). (ii) In the infinite-fluid problem, vertical oscillation of the body is represented in the far field by a distribution of vertically oriented

dipoles on the singular line. In the free-surface problem, the fluid fills only a half-space, and we cannot really distinguish between the effects of time-dependent sources at z = 0 and time-time-dependent vertical dipoles at z =0. It seems to be somewhat simpler to think in terms of sources, and we do so.

Let the source. density be given as

Re[ii(x)eil, 0 < x < L. (3.4) Then the potential is

Re{c¢(x, y, (3.5)

where (P(x, y, z) is given by [see Wehausen and Laitone (1960), Eq. (13.17")]:

L co kek.

(P(x, y, z)= _{fo th.=,e3-(0} Jo(kR) dk. (3.6)

k v

Here, R = [(x y 1/2 and v w2/g. The integral is to be interpreted

as a contour integral indented above the pole at k = v. Far away, where R, (x2 + y2)1/2 is very large, the above potential can be approximated as

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108 follows

y, z) e' 1r'cl6-(011(02)(vR) (3.7a)

2 .10

= -- evz IcxM,2)(vRo) cos nO f gEr(rc,)J(v), (3.7b)

2 0 0

where

{1,

n = 0, cc" = 2, n > 0.

The second form is obtained by applying an addition theorem for Bessel functions, as given in Eq. (9.1.79) by Abramowitz and Stegun (1964). The Hankel functions in (3.7b) can be furtherapproximated by their large-r asymptotic representations.

0 1 2 3 4

FIG. 3. Angular distribution of amplitude of radiated waves

from a line of pulsatini

sources.

We do not know 6-(x) in general; it must bedetermined somehow fromthi near-field solution. But we obtain useful insight by considering a specia case: ii(x) = G0 , a constant. This is close towhat one might find for a boxlilc ship heaving at moderate frequency. For this case, in Fig. 3 the radial

tance to the curve at any angle represents therelative magnitude of (/)for wave radiating at that same angle. Thisquantity, computed from (3.7b),

proportional to the amplitudeof the outgoing wave, and so we can tell froi 0(x,

L2

.L=5

,,L=10

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Fig. 3 whether the sourcedistribution causes waves to go out symmetrically in all directions or whether the waves are focused in certain directions. Four cases are shown. For vL = 2 (that is, )./L = in, where ;41 = 27r/v), the waves propagate outward withessentially the same amplitude in all directions. The source distributionmight just as well have beencircular, rather than rectilin-ear. At the other extreme,for vL = 20 (;t/L = n/10), the waves go out almost exclusively in the broadside direction: practically no wave energy escapesin the endwise directions nor even over a wide range of oblique angles. Two in-between cases are also shown.

The two extreme cases shown are typical results in radiation problems of acoustics, electromagnetic wavetheory, and so on. If the radiatoris small in comparison with the wavelength of the radiation, it is not possible tofocus the outgoing waves; the distributionof wave energy with direction is

essen-tially uniform. On the other hand, if the radiator is large compared with

wavelength, sharp focusing is possible. What is of interest in this example is what is meant by " large " and "small" in the comparison ofwavelength and radiator size. Apparently, a A/L ratio of about 3 is very large, and a AILratio of about 1/3 is very small.

The above results are true only veryfar away, in an asymptotic sense. It is difficult to make precise statements about the distribution of wave

ampli-tude in a region at finite distance from the oscillating ship. However, the

nearby behavior in the short-wave case is suggested qualitatively in Fig. 4. Alongside the body, there are waves propagating in the directions perpen-dicular to the axis of the body. The shorter the waves, the farther out this

wove 0 L x line ofsources----7

/

1

/

I

1

FIG. 4. Short waves generated by a line of pulsating sources.

\

--:

\

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1.10

behavior extends. Analytically, the description correspondingto Fig. 4 can

be derived from (3.7a) by using the large-R asymptotic formula for the

Hankel function and then applying the method ofstationary phase. It turns

out that there is nopoint of stationary phase unless 0 < x < L, butwithin

this range the, potential function in (3.7a) is givenapproximately as follows

4)(x, y, 43.8)

This result says that there ere waves propagating outward in the ±y direc-tions, that these waves move as if they were, strictly 2-D waves,and that the

amplitude at any point (x, y) of the free surface depends only on a(x), the

source density at the same x. Of course, the abrupt change in character of the solution along the lines x =0, L is fictitious, a result of using the method of stationary phase. If frequency is considered tobecome higher and higher (wavelength becomes shorter and shorter), the actual wave motion would

come closer and closer to this discontinuous pattern. But, for finite frequency and finite wavelength,, there must be ever-widening regions

spreading out from both endsof the ship in which this idealization is invalid. This is suggested in Fig. 4 by the broken lines emanatingfrom the body ends; in the regions bounded by these lines, the sharpdistinction becomes fuzzy between the two regions defined by the stationary-phaseprocedure.

A similar qualification must be made to the result in (3.8) that relates the wave amplitude at any(x, y) exclusively to the value of Jr(x) for the same x. This relationship becomes more nearly true closer and closer tothe singular, line.

In this discussion of radiation patterns from a heavingship, the important relationship between wavelength and frequency has been used or inferred several times, that is

2rc/.1. = v (3:9)

This is, of course, the dispersion relationship for free-surface gravity waves

on deep water. It

provides the means to rationalize possible

orderTof-,magnitude relationships between the operator alai-and the slenderness pa-rameter & [See the discussion following Eqs. (3.1) and (3.2).] For example, if we consider sinusoidal motion at radian frequency co, there is a wavelength A = 2regico2 associated with the resulting fluid motion,and, we may expect. the relationship between this and the dimensions of the ship to provide a

characteristic parameter for describing the motion. This isclear from Fig. In this way, the initially strange idea of relating the effect of time differentia-, tion to the slenderness parameter becomes quite natural: "Time"implies a frequency, which implies a wavelength, which can be compared with body dimensions from which the slenderness parameter is defined.

Furthermore suppose that we assumethat to = 0(E-1)'), so that arnt

z)eiwr iNx)evzei("-,Iyi).

= w2/g.

A

3.

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O(e 112), which is one of the possibilities mentioned in the discussion of (3.1) and (3.2). A simple plane wave at this frequency might be described in terms of a potential of the formexp(vz + hot ivy). Differentiation of this

poten-tial with respect to y or z is

equivalent in magnitude to multiplying by

v = o(c1). This is an order-of-magnitude effect not encompassed in (2.1),

and so we shall have to complement (2.1) with a formalism that adequately accounts for this effect. We shall treat this situation when we come tothe case of high-frequency oscillations.

C. FORCED OSCILLATIONS

1. Low-Frequency Oscillations

We define "low frequency" to mean that w = 0(1) as e 0, or A ---- 0(1) [see (3.9)]. We shall also refer to this as the "long-wave" problem. In prac-tice, we are likely to think of wavelength in this case as being more or less comparable to ship length, and this is a useful way of thinking. But it is not really correct and it may occasionally be misleading. The statement "A = 0(1)" implies that A/B co as e 0, where B is ship beam. The ratio A/L

may have any value, in principle, just so it remains fixed in the limit process. It could be 1/10 or even 1/100. However, the theory will not be valid unless A/B is very large, and so a small value of A/L may require that B/L be really infinitesimal. Such a theory would not be useful because no one is interested in a ship with such dimensions. So the meaning suggested when we call this a

low-frequency or long-wave case is important for practical purposes,

although it is not entirely correct.

Formally, we most often introduce the assumption co = 0(1) when we differentiate with respect to time, in which case we assume

that alai= 0(1).

Also, v and A are 0(1).

Near Field. There is assumed to be a velocity potential, (1)(x, y, z, t)=

Re[0(x, y, z)exp(iun)], to which we can apply (2.1). Then the

first

approximation for 4)(x, y, z) satisfies the 2-D Laplace equation in the near field, as in (2.6). The kinematic boundarycondition on the body is as stated in (2.7) for the infinite-fluid case. It is convenient now to redefine

U, = 0;

Uj=

j

2, 3. (3.10)

Then the body boundary condition can be rewritten:

00/ON = ico(n2 2 +

n33)

on the body. (3.11) We linearize the boundary conditions, and so we apply (3.11) on the body surface at its undisturbed position. (This is not valid in the forward-speed problem, even in the linearized case.) We assume that U; = OW; this is

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112

adequate for the present case, although we shall need a more precise state-ment later for the high-frequency case. This assumption concerning U

that

cti 0(e2), (3.12).

just as in the infinite-fluidproblems [see (2.5)].

Now we simplify the free-surface boundary conditions, (3.1) and (3.2). First consider just thelinear terms. If we keepboth linear terms in (3.1), we must expect that

;

ON)) = 0(c2), (3.13)

which then implies in (3.2) that

.= 0, (3.14)

since cb, = 0(r) [the effect of (2.1)] and ;, = 0(r2). On the other hand, we might choose to retain both linear terms in (3.2), whichwould imply that

= 0(0z) =0(e). (3.15)

Then, in (3.1), Of wouldbe of higher order than ;, and so (3.1) would reduce to

=O. (3.16)

But this is a trivial result; it means only that wedo not yet have the lowest-order term in the expansionfor ;. Therefore we reject (3.15); it appears that

(3.13) is the more reasonablechoice.

Of course, (3.14) follows directly from (3.13), and this means that the free surface has been replaced by a rigid wall in the problem that defines the first-order approximation of 4). It might seem that we have thereby

dis-carded the free surface completely, but this

is not true.

It must be

recognized that the rigid-wallcondition, (3.14), does not mean literally that

there is no vertical fluid motion at z 0; it means only that the vertical

component of fluid motion is much smaller than thehorizontal component.

From the linear terms

in

(3.1),t we can calculate the

corresponding deflection of the free surface

y, t) Re{ (11g)(1),] Re[ (ico/g)g5(x, y, 0)e"1. (3.17) From this formula, we cancalculate and we observe that it is 0(r2). The kinematic free-surface condition,

(3.2), then requires that 4): = +.

The

right-hand side is 0(0 butthe left-hand side is 0(e). If wewant to calculate

The term (1/2)4) in (3.1) is 0(1,2), the same as 4)and so it should in principlebe retained

here. However, we generally assume that we may linearize oscillationproblems with respect to

the amplitude of oscillation, in which case the quadratic termis of higher order in that sense.

im-plies

=

.

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the left-hand side to an accuracy which is only O(.), we can set the right-hand side equal to zero. And this is what (3.14) means.

The near-field problemrepresented by (3.11), (3.14), and (3.17) was

inten-sively investigated in the mid-1960s by Newman (1964), Joosen (1964),

Newman and Tuck (1964), and Maruo (1967a). This was the basis for the early slender-body theory of ship motions, which gave reasonable predic-tions at zero speed (but only at zero speed).

The solution of this near-field problem is no more difficult than in the

infinite-fluid case. Since the freesurface is replaced by a rigid wall in the 4 problem, we can extend the definition of (/) analytically intothe upper half-space as an even function with respect to z. The extended problem can be interpreted as an infinite-fluid problem, and so it is not necessary here to discuss methods of solution further.

The solution of the problemhas somewhat different properties according to whether the body oscillates vertically or horizontally. The two cases are

depicted in Fig. 5. If the body oscillates vertically, the image of the body

C:

image

(0) ( b)

FIG. 5. Motion of the body image in the low-frequency problem. (a) Vertical oscillation, (b) horizontal oscillation.

moves oppositely, so that the two act together somewhat as a pulsating body. Far away from the body,the potential can be expanded as in (2.8), and the logarithm term is the most important, that is,

4)(x,.y, z)e' {[cr(x)/271] log r + Ao(x)}e''', (3.18)

as r

(yz z2)1/2 co. [Cf. (2.8").] From conservation of mass, one can show easily that the source strength is given by

c(x)

2icoB(x)3 ,

(3.18')

where B(x) is the beam at the waterline. It may be noted that the kinetic =

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114

energy of the fluid motion per unit length of the body is infinite, afact that leads to a prediction of infinite added mass per unit length of the body.

If the body oscillateshorizontally, the image moves in the same direction,

so that the total effect is the same as for an oscillating double body in an infinite fluid. In this case then, there is no sourcelike behavior, and the

leading term in (2.8) for this problem is the horizontal dipole term, the A, term, that is,

0(x, y, z)el'r [A0(x) + A , (x) cos

0/1ei"

as r cc. (3.19) Whatever method ofsolution is used, al1. terms in the expansion (2.8) can be determined except Ao(x). There is no information available in the near field alone that can be used to determine this quantity, since the near-field boundary-value problem is a Neumann problem, the solution of which is always nonunique to the extent of an arbitrary additive constant.The

addi-tive "constant" is not trivial, since it is really a function of x, and so it

contributes to the evaluation of fluid pressure. It does notvanish, as in the

infinite-fluid case of an oscillating body; this is a consequence of the

presence of the free surface. We must solve thefar-field problem in order to determine the additive term.

Far Field. The velocity potential must satisfy the 3-D Laplace equation and the free-surfaceconditions, (3.1) and (3.2). There is no body boundary condition, only a condition that the far-field solutionbe consistent with the near-field solution. Thefar-field solution must satisfy aradiation condition,

that is, it must represent outgoing waves far awayfrom the body.

Since we are here assuming that co = 0(1) as c 0, differentiation with respect to time and space coordinates does not alterorders of magnitude, that is,

0/0t, 0/0x, 0/0y, 0/0z = 0(1) as E O. (3.20)

The potential in the far field will presumably beo(1), and so the quadratic terms in (3.1) and (3.2) are all of higher orderof magnitude than thelinear terms. It is also consistent in the first approximation tosatisfy the conditions on z = 0. Then it is asimple step to show that the free-surface conditions can be combined into the form given by (3.3), which, for sinusoidal time depen-dence, becomes

vd) Oz = 0 on z = 0, with v = w2/g. (3.21) The reasoning that followsis parallel to the infinite-fluid case: In the limit as e 0, the body shrinksdown to a line, and so the far-field solution can be

represented in terms of a distribution of singularities on that line. In the presence of the free surface, the simple singularity potentials must be

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Otherwise the distributions are arbitrary. The densities of the sources, di-poles, etc., cannot be determined solely from the far-field problem.

Furthermore, in the lowest-orderapproximation in the far field, there will

be only one kind of

singularity required and possiblewhatever kind

matches the leading-order term in the near-field solution as r oo. The

reasoning still follows that for the infinite-fluid problem. In the case of

vertical body oscillation, the near-field solution at large r behaves as in (3.18), and so the matching far-field solution represents a distribution of sources. In the case of horizontal body oscillation, the near-field solution behaves as in (3.19), and so the matchingfar-field solution represents a line of transverse dipoles.

The general problem that was discussed in connection with radiation

patterns is precisely the far-field problem for the case of a slender body undergoing vertical sinusoidal oscillations, and so the solution expressed in (3.6) can be used here directly. The estimates given in (3.7a) and (3.7b) are not useful now in setting up the matching to the near field, since they are valid only far away; however, theydo show that the solution represents outgoing waves, since

e'll2)(vR0)- (C/KI2)ei(un-s,Ro) where C is a complex constant.

In order to match (3.6) with (3.18), we must approximate the expression in (3.6) for small r. The procedure, in brief, is as follows: Rewrite (3.6) as

L oo (P(x, y, z)

-2m.

d&()dkek2.10(kR)

-o v _dk

-

_{k -}

ekzJ (kR) (3.22) 2ir v

The first term can be expressed

1 r 2&() dc

--1atx l

47r Jo [(x - )2 r2j1/2

) og r

L

- 2-7c

f

'go-10 log 2 I x -

sgn(x -

(3.23)

as r 0, that is, it represents the potential for a line distribution of sources of density 2O-(x) in an infinite fluid; the small-rapproximation for this potential was already given in (2.11)-(2.13), and we have taken it over directly here

except for the change by a factor of 2. The second term in (3.22) can be

divided into two parts, a principal-value integral and a contribution from the contour indentation; as r 0, they can be expressed in terms of standard

as RD oo,

0 .0

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116 functions

v

.10 ci -6-()[1-1,(v Ix 1) + Yo(v Ix 1) + 2iJ0(v Ix

LI;I )1, (3.24)

where Ho is a Struve function. Pius the small-r approximationfor 4)(x, y, z) is 1 4)(x, y, z) log r + f (x), where f (x) 47c Jo dditv[110(vIx )) + Y0(v1x )) + 200(1'1x 1)]

2 log 21x

sgn(x (3.25')

After being multiplied by exp(ian), this must match with (3.18), and so we have

6-(x) = icoB(x)3; (3.26a)

A0(x) =f (x). (3.26b)

Just as in the infinite-fluid problem, the near field hasprovided the infor-mation that was lacking in thefar field, namely, the value of ii(x), and the far field has provided theinformation that was lacking in the nearfield, namely, the additive function Ao(x). The latter, in particular, isessential information, for this " constant " in the solution of the 2-D crossplaneproblems contrib-utes to the pressure field acting on the body. This extra pressure does not depend on y and z, but it maynevertheless cause a net force in the transverse directions, since the pressure is integrated over just thewetted portion of the hull to give net force. In any case, the term A0(x) in the near-fieldsolution i; the only manifestation of interactions among the crosssections at variow values of x; it represents a longitudinal wave at a particular xcaused by au sources (in the far-field description) at all other values of x. If A0(x) wen equal to zero, the above slender-ship theory would be just a primitive kind o

strip theory.

In the case of horizontal body oscillations, the resultingtheory is just primitive strip theory, with no interactions among cross sections. The situa tion is analogous to the infinite-fluid case. The additive function of x canno

be determined in the near field alone; all that can be asserted from th

near-field problem is that, for large r, the solution behaves like a constanto like cos 0/r [see (3.19)]. This fact eliminates the possibility of there being

(3.25)

L

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line of sources in the lowest-order far-field solution; the first term must

represent a line distribution of horizontal dipoles of unknown density. The potential for such a distribution can be written down immediately, for exam-ple, by differentiating (3.22) with respect to y. This potential can be approx-imated for small r by a single-term expression that represents a 2-Ddipole, that is, it matches the A, term of (3.19), and there is nothing to match with A0. Therefore we must set Ao ---- 0 in the near-field solution for the case of horizontal body oscillations, and there are no interaction effects among the

various cross sections.

Nonuniformity of the Near-Field Solution. Slender-body theory isby its nature a study in singular-perturbation problems, marked by strong math-ematical nonuniformities in the solutions. The far-field solution is nonuni-form near the singular line. The near-field solution is nonuninonuni-form far away, for

it gives no solution (or a

trivial solution) outside of the domain

0 < x <L The low-frequencyship-oscillation case presents an even

stron-ger nonuniformity in the near-field solution: It admits of no wavelike motion [except for the add-on longitudinal wave component included in Ap(x)]. Physically, one expects a heaving ship to create outgoing waves and these waves ought to be evident in the regions off the sides of the ship. In fact, this is what Fig. 3 showed on the basis of a simple mathematical model. However, the near-field solution displays no such behavior even as y --0 + co in the near field.

The interpretation of this fact is that the wavelength is so very, very long

that the behavior identified as "near-field behavior" occurs entirely in a region which is small compared with a wavelength. Of course, we have

already described this as a " long-wave theory," and we implied that ship beam was small compared with the wavelength. But much more is implied: the whole near field is small compared with the wavelength. This seems tobe a rather severe constraint onthe possible utility of the theory. It also implies

that there is no damping (to lowest order) associated with the waves

ra-diated out to the sides.

Another aspect of possibly singular behavior may be noted, this being a local singularity. In the case of vertical oscillation of the body, the fluid velocity at the level z = 0 on the side of the body can have only a vertical component. [We are assuming that the body is wall-sided at thewaterline.] However, the rigid-wall condition at the free surface says that there can be only a horizontal component of velocity at that samepoint. Accordingly, the solution must indicate a stagnation point atthe juncture of the body surface and the undisturbed free surface. Such a flow condition at that point seems highly unlikely, however. Our general interpretation of the rigid-wall condi-tion seems to be valid enough: The vertical velocity component is very small compared with the horizontal component. However, at the body side the

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latter is precisely zero in magnitude, and so this comparison ismeaningless there. Our assumptions have led us into setting a poorly posed problem

in

this respect, but we do not know of anycareful analysis of this point. 2. High-Frequency Oscillations: "Strip Theory"

"High frequency" will be defined by the order-of-magnitude statement w 0(z-112). The physicalinterpretation of this should

be made in termsof the corresponding wavelength, /1 = 27Egy'w2, which

is now 0(c). Since

B = OW as well, we presumethat the ratio Al B is 0(1)as E 0. Thus we are assuming that the frequency becomes higher and higher as the body be-comes more slender, and it does so in just such a way that A/13 can remain fixed. This may seem to be highly artificial, but in fact it leads to an ex-tremely useful theory. As a heuristic explanation, we may say that it is important under certain circumstances to relatewavelength primarily to the ship beam (or possibly draft), rather than to shiplength, as was done in the

foregoing low-frequency case.

The resulting theorycorresponds generally to what is usually called "strip theory" in naval architecture. The ordinary derivations of this striptheory are quitedifferent from what we shall presenthere, although it haslong been recognized that "high frequency" is generally implied.

In the current literature onship hydrodynamics, we sometimes encounter an objection being made to the use of a high-frequency theory, usually on the basis of the argument that the wavelength is notmuch smaller than ship length in the most important problems of wave loadsand ship motions. To some extent, such an objection misses the practical

significance of asymp-totic solutions: We develop an asymptotic expansion in terms of a small parameter which is supposed to be approaching zero,and then we apply the theory to cases in whichthe small parameter hasquite a finite value. We can seldom tell from the theory alone whether the finitevalue of the parameter is really small enough for the approximation to be useful; we never seem to know how small is "small." In one problem,the small parametermight have to be 10-6, and in another problem it might be unity. We musthave some other basis for judging the validity of the approximation, perhaps

experi-ments, a higher-order calculation, or a special case that can be solved

exactly. Experiments, in particular, have demonstrated that theassumptior of high frequency in the ship-motion problem gives more accurateanswer! than the assumption of low frequency.

Near Field. The assumption that co = 0(c-112) arises naturally from consideration of the near-field problem. If we combinethe two free-surfao conditions, (3.1) and (3.2), and retain only thelinear terms, we obtain (3.3; that is,

4) + g4 = 0

on z 0. (3.27

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In the near field,

ctiz =

If the motion is sinusoidal at

radian frequency 0), then = 0(00)2). There areobviously only three possibilities to consider: (i) W = o(E-1/2); (ii) =-- 0(E-112) [and w-1 = 0(c1/2)1; (iii)

0-1 = o(012). The first case led to the low-frequency problem of the last

section, with the rigid-wall free-surface condition. The second case is the present one, and we seethat both terms in thefree-surface condition must be retained; they are strictly of the same order of magnitude.In the third case, the frequency is so high that only the first term in thefree-surface condition is retained, that is, cf) = 0, which is true for all time,and so 0 on z = 0.

It should be observed that only case (ii) above allows directly for the

occurrence of gravity waves in the near field, and so it is in many ways the most interesting case. This is the problem that we now consider.

The argument above is based on consideration of just the linear termsin the free-surface conditions, (3.1) and (3.2). In problemsof small oscillations of a body in the free surface, we generally expect that it will be legitimate to linearize the conditions, and so we expect that the conclusions above will stand when the nonlinear terms areconsidered too. This formal linearization will be carried out in an explicit wayin Section V. where we encounter more difficulties if we simply try tolinearize by inspection. For the present,we do not need that degree of formalism. However, the consequences of lineariza-tion should be noted: We lose all harmonic responses and the occasionally important steady (mean) force.

The body boundary condition is exactly as in the low-frequencyproblem, Eq. (3.11). We can even include pitch and yaw motions I in the formulation by defining as the amplitude of pitch (rotation about the yaxis) and c6 as the amplitude of yaw (rotation about the z axis). These angular displace-ments are assumed to be small enough to be treated as components of a rotation vector. A generalized form of ni is also required: For j = 5, 6, we define

e_3 (r x n),

(3.28)

where ei is the unit vector parallel to the x; axis, r is the position vector of a

point on the body surface, and n = (n1, n2, n3) is the usual unit normal

Note that, without this alternative statement for case (ii), we should have to accept the situation that case (ii) encompasses case (i), whereas we want them to be mutually exclusive.

The reader is reminded that the statement y = 0(x) as x 0 means that y/x I remains

bounded in the limit. The statement y = o(x) means that I y/x 0. If y and x satisfy the latter statement, then they also satisfy the former.

A similar extension is, of course, possible for roll and surge motions, but the resulting theory is of doubtful validity. The question about surge was mentioned atthe beginning of

Section III,A. With respect to roll motion, the difficulty probably lies more in theassumption of

an ideal fluid. In practical situations, large amplitude may also invalidate the theory.

parallel to the ship

axis, and so the assumption about lex in (2.1) is

retained. This assumptionshould be a very good one over most of the length of the ship, but we must recognize that it cannot really be true near the

ship ends, and so some error is thereby introduced.The higher the frequency, the better is this assumption.

The short-wave nature of the solution does not introduce any complica-tion with respect to the assumpcomplica-tion in (2.1) about derivatives in the trans-verse directions. We continue

to assume that

Olily and 010z have

order-of-magnitude effects like multiplying by

c-'; it does not matter

whether this is a result of slenderness or ofthe short-wave character of the

fluid motion.

So finally we are free to use the 2-D Laplaceequation in the near field. The

solution must satisfy the body boundary condition (3.29) and the

free-surface condition (3.30). We expect the solution to represent outgoing waves at large 1.1' 1, but amore basicapproach is to require that the solution should match properly with a far-field outgoing-wavesolution.

This 2-D problem has been studiedextensively over a period of 30 years,